Lectures co-financed by the European Union in scope of the European Social Fund

1. Crystallography and Diffraction Theory and Modern Methods of Analysis Lectures 3-4Lattice Planes and the Reciprocal LatticeDr. I. AbrahamsQueen Mary University of London Lectures co-financed by the European Union in scope of the European Social Fund

2. Lattice Planes Early in the history of crystallography and diffraction WL Bragg showed that diffraction of X-rays by crystals could be explained in terms of reflections from semi-transparent mirrors or planes within the crystal lattice. These planes are termed Lattice Planes. Consider a 2D-lattice On the right hand side a set of parallel lines have been drawn on this 2D-lattice. How do we distinguish this set of lines from other sets of lines ? If we consider the spaces between lattice points, then travelling along vector a two spaces are crossed between lattice points and along b one space is crossed. We can therefore index this set of lines as 2,1 Lectures co-financed by the European Union in scope of the European Social Fund

3. Now consider the following lines on the same 2D-lattice 2,1 or In these cases, travelling along the axial vectors between lattice points, the same number of spaces are crossed. However, whereas in the first case the lines are crossed from the same side, in the second case travelling along a, lines are crossed on the reverse side to that when travelling along b. Hence, one of the indices is given a negative value. In this example, the same lattice can be described by either a primitive or centred unit cell. The set of lattice lines shown is given different indices for the two cells. For the primitive cell the lines are indexed as 2,1 while for the centred cell the same lines are indexed as 2,0. Lectures co-financed by the European Union in scope of the European Social Fund

4. The same rules apply to 3D-lattices which require three indices designated h, k, l These are known as the Miller indices, where h is the index corresponding to the a-axis k is the index corresponding to the b-axis l is the index corresponding to the c-axis The directions of the lattice vectors are normally chosen according to the right hand rule. i.e. the thumb, first finger and second finger correspond to +x (a), +y (b) and +z (c) directions respectively. Right hand convention for crystal lattices. Ref: X-ray Structure Determination. A practical Guide. G.H. Stout and L.H. jensen 2nd ed. 1989 Wiley New York. Lectures co-financed by the European Union in scope of the European Social Fund

5. Lectures co-financed by the European Union in scope of the European Social Fund

6. When trying to sketch lattice planes for more complicated cases it is best to first mark out the cell edges. Then, starting from the marks nearest to the origin join the marks Continue joining up the marks until all are used. Remember what you are drawing are parallel planes in 3D. Lectures co-financed by the European Union in scope of the European Social Fund

7. Finally, shade the planes to make them clearer Lectures co-financed by the European Union in scope of the European Social Fund

8. d-Spacings The perpendicular distance between parallel planes is known as the d-spacing and in normally quoted in Å e.g. 0,2,0 planes in an orthorhombic cell d020 = b/2 For a simple orthogonal system the d-spacing is easily calculated where two of the Miller indices are zero. Generally for an orthogonal crystal system (i.e. orthorhombic, tetragonal or cubic) Lectures co-financed by the European Union in scope of the European Social Fund

9. e.g. The following diffraction peaks were collected for a tetragonal cell using Cu-K radiation ( = 1.5418 Å). Calculate the unit cell parameters a and c. h k l 2 / 2 0 0 40.073 1 1 1 30.942 First use Bragg's Law  = 2 dhkl sinhkl to convert the 2 values to d-spacings in Å. i.e Å Å Since the unit cell is tetragonal (i.e. the axes are orthogonal) a = b = 2 d200 = 4.500 Å. Now we know a and b we can use the 111 reflection to calculate c. c = 6.906 Å Lectures co-financed by the European Union in scope of the European Social Fund

10. Direct Cell Equations for d-Spacings Triclinic where Monoclinic Orthorhombic Tetragonal Hexagonal Cubic Lectures co-financed by the European Union in scope of the European Social Fund Lectures co-financed by the European Union in scope of the European Social Fund

11. The Reciprocal Lattice The reciprocal lattice is a theoretical concept that makes the interpretation of X-ray diffraction data easier. Consider a 2D optical diffraction grating a If we shine light through this grating we get a diffraction pattern b Lectures co-financed by the European Union in scope of the European Social Fund

12. These spots can be indexed as follows: So 0 2 is twice as far away from the centre as 0 1 etc. The spacings of the points are inversely proportional to the lattice spacings. X-ray patterns are analogous in 3D. Interpretation of these patterns is easier if we redefine the lattice in terms of the direction and spacings of the lattice planes This new lattice is called the Reciprocal Lattice. Lectures co-financed by the European Union in scope of the European Social Fund

13. Consider a simple orthogonal lattice. A projection down the c-axis of the unit cell is shown below. Starting at the origin in the real (direct) lattice, lines are drawn perpendicular to the lattice planes.010 and 100 These lines are marked at points d* where d* = 1/d. Using this convention, d* has units Å-1. However, it is more convenient to use d* = /d which has dimensionless units known as reciprocal lattice units (r.l.u.). Lectures co-financed by the European Union in scope of the European Social Fund

14. This is continued to give a layer of the reciprocal lattice Note that because in this case the direct lattice is orthogonal a* is parallel to a and b* is parallel to b. The magnitude of a* is inversely proportional to a and similarly b* to b. Additional layers of the reciprocal lattice can be built up in the c* direction to give a sphere of reciprocal space. Lectures co-financed by the European Union in scope of the European Social Fund

15. Consider now the case of a non-orthogonal direct lattice such as in a monoclinic system. • a = 6 Å • c = 4 Å • = 110 (b = 10 Å) Therefore a* = 0.177 c* = 0.266 * = 70 Lectures co-financed by the European Union in scope of the European Social Fund

16. Triclinic symmetry represents the general case: Lectures co-financed by the European Union in scope of the European Social Fund

17. If we assign an intensity to each reciprocal lattice point corresponding to the intensity of the observed X-ray reflection then we obtain an intensity weighted reciprocal lattice. Lectures co-financed by the European Union in scope of the European Social Fund

18. The diffraction pattern shown by the weighted reciprocal lattice will show the point symmetry and the centre of symmetry of the crystal structure (if it is centrosymmetric). This is known as Laue symmetry. Simple X-ray photographs can be taken of the undistorted reciprocal lattice using a variety of cameras. Max von Laue de Jong Bouman photograph of the hk0 layer of a crystal of ammonium oxalate monohydrate (space group P21212) Lectures co-financed by the European Union in scope of the European Social Fund